# The size of an intertwine

## Abstract

An intertwine of two graphs *H* and *H′* is a graph *G* such that *G* contains both *H* and *H′* as minors, but no proper minor of *G* contains both *H* and *H′* as minors. We give an upper bound on the size of an intertwine of two given planar graphs. For two planar graphs *H* and *H′*, the bound is triply exponential in *O(m*^{5}) where *m*≤max(¦*V(H)*¦, ¦*V(H*′)¦). We also give an upper on the size of an intertwine of two given trees *T* and *T′*. This bound is exponential in *O(m*^{3} log *m*) where m≤max(¦*V(T*)¦, ¦*V(T*′)¦). Let *O*_{1} be the set of obstructions for a minor closed family *L*_{1} and *O*_{2} the set of obstructions for a minor closed family *L*_{2}. It is a natural to ask the following question: how do we given *O*_{1} and *O*_{2} compute the obstructions for *L*_{1}∩*L*_{2} and *L*_{1} ∪ *L*_{2}. Both these sets of obstructions are known to be finite, and to obtain the obstructions for *L*_{1}∩ *L*_{2} is easy. However, to compute the obstructions for *L*_{1}*∪ L*_{2} is hard. Our upper bound enables us to given *O*_{1} and *O*_{2} compute a bound on the size of any obstruction for *L*_{1} ∪ *L*_{2} whenever *L*_{1} and *L*_{2} are families of planar graphs.

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